Constructive Interference Problem in the Time Domain

1. The problem statement, all variables and given/known data
Two waves on a string are given by the following functions:
Y1 (x,t) = 4cos(20t-x)
Y2 (x,t) = -4cos(20t+x)
where x is in centimeters. The waves are said to interfere constructively when their superposition |Ys| = |Y1 + Y2| is a maximum and they interfere destructively when |Ys|
is a minimum.

if t = ∏/50 seconds, at what location x is the interference constructive?

2. Relevant equations
No particular equation relevant as far as I know.

3. The attempt at a solution
So to get a constructive interference, the summation of the two waves(|Y1 + Y2|) must be the largest possible. I plugged in the value for time, and got this simplified equation for Ys:

Ys = |4[cos(2∏/5 - x) - cos(2∏/5 + x)]|

Now i know |Ys| must be the largest it can be, and the only way i can think of of approaching this is constructing a X-Y table and seeing if there is a trend in the values, though I feel there must be an easier way to do this.

Staff: Mentor

1. The problem statement, all variables and given/known data
Two waves on a string are given by the following functions:
Y1 (x,t) = 4cos(20t-x)
Y2 (x,t) = -4cos(20t+x)
where x is in centimeters. The waves are said to interfere constructively when their superposition |Ys| = |Y1 + Y2| is a maximum and they interfere destructively when |Ys|
is a minimum.

if t = ∏/50 seconds, at what location x is the interference constructive?

2. Relevant equations
No particular equation relevant as far as I know.

3. The attempt at a solution
So to get a constructive interference, the summation of the two waves(|Y1 + Y2|) must be the largest possible. I plugged in the value for time, and got this simplified equation for Ys:

Ys = |4[cos(2∏/5 - x) - cos(2∏/5 + x)]|

Now i know |Ys| must be the largest it can be, and the only way i can think of of approaching this is constructing a X-Y table and seeing if there is a trend in the values, though I feel there must be an easier way to do this.

Yes, there is an easier way. If you add two sinusoidal functions that are in phase, what is the maximum amplitude that you can get?